MBCNS2 Example Book

MBCNS2 is a small collection of programs for the simulation of transient two-dimensional (or axisymmetric) flows. It is part of the larger collection of compressible flow simulation codes found at http://www.mech.uq.edu.au/cfcfd/. This manual is a collection of example simulations: scripts, results and commentary. It may be convenient for new users of the code to identify an example close to the situation that they wish to model and then adapt the scripts for that example.

Theory of multidimensional parametric band-gap simultons

Multidimensional spatiotemporal parametric simultons (simultaneous solitary waves) are possible in a nonlinear chi((2)) medium with a Bragg grating structure, where large effective dispersion occurs near two resonant band gaps for the carrier and second-harmonic field, respectively. The enhanced dispersion allows much reduced interaction lengths, as compared to bulk medium parametric simultons. The nonlinear parametric band-gap medium permits higher-dimensional stationary waves to form. In addition, solitons can occur with lower input powers than conventional nonlinear Schrodinger equation gap solitons. In this paper, the equations for electromagnetic propagation in a grating structure with a parametric nonlinearity are derived from Maxwell's equation using a coupled mode Hamiltonian analysis in one, two, and three spatial dimensions. Simultaneous solitary wave solutions are proved to exist by reducing the equations to the coupled equations describing a nonlinear parametric waveguide, using the effective-mass approximation (EMA). Exact one-dimensional numerical solutions in agreement with the EMA solutions are also given. Direct numerical simulations show that the solutions have similar types of stability properties to the bulk case, providing the carrier waves are tuned to the two Bragg resonances, and the pulses have a width in frequency space less than the band gap. In summary, these equations describe a physically accessible localized nonlinear wave that is stable in up to 3 + 1 dimensions. Possible applications include photonic logic and switching devices. [S1063-651X(98)06109-1].

Dynamical quantum noise in trapped Bose-Einstein condensates

We introduce the study of dynamical quantum noise in Bose-Einstein condensates through numerical simulation of stochastic partial differential equations obtained using phase-space representations. We derive evolution equations for a single trapped condensate in both the positive-P and Wigner representations and perform simulations to compare the predictions of the two methods. The positive-P approach is found to be highly susceptible to the stability problems that have been observed in other strongly nonlinear, weakly damped systems. Using the Wigner representation, we examine the evolution of several quantities of interest using from a variety of choices of initial stare for the condensate and compare results to those for single-mode models. [S1050-2947(98)06612-8].

Evolution of three-dimensional folds for a non-Newtonian plate in a viscous medium

We derive analytical solutions for the three-dimensional time-dependent buckling of a non-Newtonian viscous plate in a less viscous medium. For the plate we assume a power-law rheology. The principal, axes of the stretching D-ij in the homogeneously deformed ground state are parallel and orthogonal to the bounding surfaces of the plate in the flat state. In the model formulation the action of the less viscous medium is replaced by equivalent reaction forces. The reaction forces are assumed to be parallel to the normal vector of the deformed plate surfaces. As a consequence, the buckling process is driven by the differences between the in-plane stresses and out of plane stress, and not by the in-plane stresses alone as assumed in previous models. The governing differential equation is essentially an orthotropic plate equation for rate dependent material, under biaxial pre-stress, supported by a viscous medium. The differential problem is solved by means of Fourier transformation and largest growth coefficients and corresponding wavenumbers are evaluated. We discuss in detail fold evolutions for isotropic in-plane stretching (D-11 = D-22), uniaxial plane straining (D-22 = 0) and in-plane flattening (D-11 = -2D(22)). Three-dimensional plots illustrate the stages of fold evolution for random initial perturbations or initial embryonic folds with axes non-parallel to the maximum compression axis. For all situations, one dominant set of folds develops normal to D-11, although the dominant wavelength differs from the Biot dominant wavelength except when the plate has a purely Newtonian viscosity. However, in the direction parallel to D-22, there exist infinitely many modes in the vicinity of the dominant wavelength which grow only marginally slower than the one corresponding to the dominant wavelength. This means that, except for very special initial conditions, the appearance of a three-dimensional fold will always be governed by at least two wavelengths. The wavelength in the direction parallel to D-11 is the dominant wavelength, and the wavelength(s) in the direction parallel to D-22 is determined essentially by the statistics of the initial state. A comparable sensitivity to the initial geometry does not exist in the classic two-dimensional folding models. In conformity with tradition we have applied Kirchhoff's hypothesis to constrain the cross-sectional rotations of the plate. We investigate the validity of this hypothesis within the framework of Reissner's plate theory. We also include a discussion of the effects of adding elasticity into the constitutive relations and show that there exist critical ratios of the relaxation times of the plate and the embedding medium for which two dominant wavelengths develop, one at ca. 2.5 of the classical Biot dominant wavelength and the other at ca. 0.45 of this wavelength. We propose that herein lies the origin of parasitic folds well known in natural examples.

Algebraic Bethe ansatz for integrable Kondo impurities in the one-dimensional supersymmetric t-J model

An integrable Kondo problem in the one-dimensional supersymmetric t-J model is studied by means of the boundary supersymmetric quantum inverse scattering method. The boundary K matrices depending on the local moments of the impurities are presented as a nontrivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained. (C) 1999 Elsevier Science B.V.

Integrable Kondo impurities in the one-dimensional supersymmetric extended Hubbard model

An integrable Kondo problem in the one-dimensional supersymmetric extended Hubbard model is studied by means of the boundary graded quantum inverse scattering method. The boundary K-matrices depending on the local moments of the impurities are presented as a non-trivial realization of the graded reflection equation algebras in a two-dimensional impurity Hilbert space. Further, the model is solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.

Spatiotemporal solitons in multidimensional optical media with a quadratic nonlinearity

We consider solutions to the second-harmonic generation equations in two-and three-dimensional dispersive media in the form of solitons localized in space and time. As is known, collapse does not take place in these models, which is why the solitons may be stable. The general solution is obtained in an approximate analytical form by means of a variational approach, which also allows the stability of the solutions to be predicted. Then, we directly simulate the two-dimensional case, taking the initial configuration as suggested by the variational approximation. We thus demonstrate that spatiotemporal solitons indeed exist and are stable. Furthermore, they are not, in the general case, equivalent to the previously known cylindrical spatial solitons. Direct simulations generate solitons with some internal oscillations. However, these oscillations neither grow nor do they exhibit any significant radiative damping. Numerical solutions of the stationary version of the equations produce the same solitons in their unperturbed form, i.e., without internal oscillations. Strictly stable solitons exist only if the system has anomalous dispersion at both the fundamental harmonic and second harmonic (SH), including the case of zero dispersion at SH. Quasistationary solitons, decaying extremely slowly into radiation, are found in the presence of weak normal dispersion at the second-harmonic frequency.

Three-dimensional structure of RTD-1, a cyclic antimicrobial defensin from rhesus macaque leukocytes

Most mammalian defensins are cationic peptides of 29-42 amino acids long, stabilized by three disulfide bonds. However, recently Tang et al. (1999, Science 286, 498-502) reported the isolation of a new defensin type found in the leukocytes of rhesus macaques. In contrast to all the other defensins found so far, rhesus theta defensin-1 (RTD-1) is composed of just 18 amino acids with the backbone cyclized through peptide bonds. Antibacterial activities of both the native cyclic peptide and a linear form were examined, showing that the cyclic form was 3-fold more active than the open chain analogue [Tang et al. (1999) Science 286, 498-502]. To elucidate the three-dimensional structure of RTD-1 and its open chain analogue, both peptides were synthesized using solid-phase peptide synthesis and tert-butyloxycarbonyl chemistry. The structures of both peptides in aqueous solution were determined from two-dimensional H-1 NMR data recorded at 500 and 750 MHz. Structural constraints consisting of interproton distances and dihedral angles were used as input for simulated-annealing calculations and water refinement with the program CNS. RTD-1 and its open chain analogue oRTD-1 adopt very similar structures in water. Both comprise an extended beta -hairpin structure with turns at one or both ends. The turns are well defined within themselves and seem to be flexible with respect to the extended regions of the molecules. Although the two strands of the beta -sheet are connected by three disulfide bonds, this region displays a degree of flexibility. The structural similarity of RTD-1 and its open chain analogue oRTD-1, as well as their comparable degree of flexibility, support the theory that the additional charges at the termini of the open chain analogue rather than overall differences in structure or flexibility are the cause for oRTD-1's lower antimicrobial activity. In contrast to numerous other antimicrobial peptides, RTD-1 does not display any amphiphilic character, even though surface models of RTD-1 exhibit a certain clustering of positive charges. Some amide protons of RTD-1 that should be solvent-exposed in monomeric beta -sheet structures show low-temperature coefficients, suggesting the possible presence of weak intermolecular hydrogen bonds.

Computation of the linear elastic properties of random porous materials with a wide variety of microstructure

A finite-element method is used to study the elastic properties of random three-dimensional porous materials with highly interconnected pores. We show that Young's modulus, E, is practically independent of Poisson's ratio of the solid phase, nu(s), over the entire solid fraction range, and Poisson's ratio, nu, becomes independent of nu(s) as the percolation threshold is approached. We represent this behaviour of nu in a flow diagram. This interesting but approximate behaviour is very similar to the exactly known behaviour in two-dimensional porous materials. In addition, the behaviour of nu versus nu(s) appears to imply that information in the dilute porosity limit can affect behaviour in the percolation threshold limit. We summarize the finite-element results in terms of simple structure-property relations, instead of tables of data, to make it easier to apply the computational results. Without using accurate numerical computations, one is limited to various effective medium theories and rigorous approximations like bounds and expansions. The accuracy of these equations is unknown for general porous media. To verify a particular theory it is important to check that it predicts both isotropic elastic moduli, i.e. prediction of Young's modulus alone is necessary but not sufficient. The subtleties of Poisson's ratio behaviour actually provide a very effective method for showing differences between the theories and demonstrating their ranges of validity. We find that for moderate- to high-porosity materials, none of the analytical theories is accurate and, at present, numerical techniques must be relied upon.

Turbulent flow around a rotating stepped cylinder

Direct numerical simulation (DNS) of turbulent flow around a rotating cylinder with two backward-facing steps axisymmetrically mounted in the circumferential direction was performed and compared with DNS of plane backward-facing step flow (PBSF) of Le [J. Fluid Mech. 330, 349 (1997)]. The original motivation of this work stemmed from the efforts to design a simple device which can generate flows of high turbulence intensity at low cost for corrosion researchers. It turned out that the current flow shows flow structures quite similar to those of PBSF downstream of the step, even though configurations of the two flows are totally different from one another. The stepped cylinder appears to be a cost-effective tool in the generation of flow structures similar to those of PBSF. (C) 2002 American Institute of Physics.

Polyethylene flow prediction with a differential multi-mode Pom-Pom model

We report the first steps of a collaborative project between the University of Queensland, Polyflow, Michelin, SK Chemicals, and RMIT University; on simulation, validation and application of a recently introduced constitutive model designed to describe branched polymers. Whereas much progress has been made on predicting the complex flow behaviour of many - in particular linear - polymers, it sometimes appears difficult to predict simultaneously shear thinning and extensional strain hardening behaviour using traditional constitutive models. Recently a new viscoelastic model based on molecular topology, was proposed by McLeish and Larson (1998). We explore the predictive power of a differential multi-mode version of the pom-pom model for the flow behaviour of two commercial polymer melts: a (long-chain branched) low-density polyethylene (LDPE) and a (linear) high-density polyethylene (HDPE). The model responses are compared to elongational recovery experiments published by Langouche and Debbaut (1999), and start-up of simple shear flow, stress relaxation after simple and reverse step strain experiments carried out in our laboratory.

Simulation of the load-unload response ratio and critical sensitivity in the lattice solid model

The Load-Unload Response Ratio (LURR) method is an intermediate-term earthquake prediction approach that has shown considerable promise. It involves calculating the ratio of a specified energy release measure during loading and unloading where loading and unloading periods are determined from the earth tide induced perturbations in the Coulomb Failure Stress on optimally oriented faults. In the lead-up to large earthquakes, high LURR values are frequently observed a few months or years prior to the event. These signals may have a similar origin to the observed accelerating seismic moment release (AMR) prior to many large earthquakes or may be due to critical sensitivity of the crust when a large earthquake is imminent. As a first step towards studying the underlying physical mechanism for the LURR observations, numerical studies are conducted using the particle based lattice solid model (LSM) to determine whether LURR observations can be reproduced. The model is initialized as a heterogeneous 2-D block made up of random-sized particles bonded by elastic-brittle links. The system is subjected to uniaxial compression from rigid driving plates on the upper and lower edges of the model. Experiments are conducted using both strain and stress control to load the plates. A sinusoidal stress perturbation is added to the gradual compressional loading to simulate loading and unloading cycles and LURR is calculated. The results reproduce signals similar to those observed in earthquake prediction practice with a high LURR value followed by a sudden drop prior to macroscopic failure of the sample. The results suggest that LURR provides a good predictor for catastrophic failure in elastic-brittle systems and motivate further research to study the underlying physical mechanisms and statistical properties of high LURR values. The results provide encouragement for earthquake prediction research and the use of advanced simulation models to probe the physics of earthquakes.

Comparison of DEM and experiment for a scale model SAG mill

Predictions of flow patterns in a 600-mm scale model SAG mill made using four classes of discrete element method (DEM) models are compared to experimental photographs. The accuracy of the various models is assessed using quantitative data on shoulder, toe and vortex center positions taken from ensembles of both experimental and simulation results. These detailed comparisons reveal the strengths and weaknesses of the various models for simulating mills and allow the effect of different modelling assumptions to be quantitatively evaluated. In particular, very close agreement is demonstrated between the full 3D model (including the end wall effects) and the experiments. It is also demonstrated that the traditional two-dimensional circular particle DEM model under-predicts the shoulder, toe and vortex center positions and the power draw by around 10 degrees. The effect of particle shape and the dimensionality of the model are also assessed, with particle shape predominantly affecting the shoulder position while the dimensionality of the model affects mainly the toe position. Crown Copyright (C) 2003 Published by Elsevier Science B.V. All rights reserved.

Comparação de desempenho entre a formulação direta do Método dos Elementos de Contorno com Funções Radiais e o Método dos Elementos Finitos em problemas de Poisson e Helmholtz

O presente trabalho objetiva avaliar o desempenho do MECID (Método dos Elementos de Contorno com Interpolação Direta) para resolver o termo integral referente à inércia na Equação de Helmholtz e, deste modo, permitir a modelagem do Problema de Autovalor assim como calcular as frequências naturais, comparando-o com os resultados obtidos pelo MEF (Método dos Elementos Finitos), gerado pela Formulação Clássica de Galerkin. Em primeira instância, serão abordados alguns problemas governados pela equação de Poisson, possibilitando iniciar a comparação de desempenho entre os métodos numéricos aqui abordados. Os problemas resolvidos se aplicam em diferentes e importantes áreas da engenharia, como na transmissão de calor, no eletromagnetismo e em problemas elásticos particulares. Em termos numéricos, sabe-se das dificuldades existentes na aproximação precisa de distribuições mais complexas de cargas, fontes ou sorvedouros no interior do domínio para qualquer técnica de contorno. No entanto, este trabalho mostra que, apesar de tais dificuldades, o desempenho do Método dos Elementos de Contorno é superior, tanto no cálculo da variável básica, quanto na sua derivada. Para tanto, são resolvidos problemas bidimensionais referentes a membranas elásticas, esforços em barras devido ao peso próprio e problemas de determinação de frequências naturais em problemas acústicos em domínios fechados, dentre outros apresentados, utilizando malhas com diferentes graus de refinamento, além de elementos lineares com funções de bases radiais para o MECID e funções base de interpolação polinomial de grau (um) para o MEF. São geradas curvas de desempenho através do cálculo do erro médio percentual para cada malha, demonstrando a convergência e a precisão de cada método. Os resultados também são comparados com as soluções analíticas, quando disponíveis, para cada exemplo resolvido neste trabalho.

Identification of tsunami-induced deposits using numerical modeling and rock magnetism techniques: A study case of the 1755 Lisbon tsunami in Algarve, Portugal

Storm- and tsunami-deposits are generated by similar depositional mechanisms making their discrimination hard to establish using classic sedimentologic methods. Here we propose an original approach to identify tsunami-induced deposits by combining numerical simulation and rock magnetism. To test our method, we investigate the tsunami deposit of the Boca do Rio estuary generated by the 1755 earthquake in Lisbon which is well described in the literature. We first test the 1755 tsunami scenario using a numerical inundation model to provide physical parameters for the tsunami wave. Then we use concentration (MS. SIRM) and grain size (chi(ARM), ARM, B1/2, ARM/SIRM) sensitive magnetic proxies coupled with SEM microscopy to unravel the magnetic mineralogy of the tsunami-induced deposit and its associated depositional mechanisms. In order to study the connection between the tsunami deposit and the different sedimentologic units present in the estuary, magnetic data were processed by multivariate statistical analyses. Our numerical simulation show a large inundation of the estuary with flow depths varying from 0.5 to 6 m and run up of similar to 7 m. Magnetic data show a dominance of paramagnetic minerals (quartz) mixed with lesser amount of ferromagnetic minerals, namely titanomagnetite and titanohematite both of a detrital origin and reworked from the underlying units. Multivariate statistical analyses indicate a better connection between the tsunami-induced deposit and a mixture of Units C and D. All these results point to a scenario where the energy released by the tsunami wave was strong enough to overtop and erode important amount of sand from the littoral dune and mixed it with reworked materials from underlying layers at least 1 m in depth. The method tested here represents an original and promising tool to identify tsunami-induced deposits in similar embayed beach environments.